# course4_1103 - Course 4 Fall 2003 Society of...

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Course 4: Fall 2003 -1- GO ON TO NEXT PAGE Fall 2003 Society of Actuaries **BEGINNING OF EXAMINATION** 1. You are given the following information about a stationary AR(2) model: (i) ρ 1 05 = . (ii) 2 01 = Determine φ 2 . (A) –0.2 (B) 0.1 (C) 0.4 (D) 0.7 (E) 1.0

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Course 4: Fall 2003 -2- GO ON TO NEXT PAGE 2. You are given: (i) Losses follow a loglogistic distribution with cumulative distribution function: Fx x x bg = + / / θ γ 1 (ii) The sample of losses is: 10 35 80 86 90 120 158 180 200 210 1500 Calculate the estimate of by percentile matching, using the 40 th and 80 th empirically smoothed percentile estimates. (A) Less than 77 (B) At least 77, but less than 87 (C) At least 87, but less than 97 (D) At least 97, but less than 107 (E) At least 107
Course 4: Fall 2003 -3- GO ON TO NEXT PAGE 3. You are given: (i) The number of claims has a Poisson distribution. (ii) Claim sizes have a Pareto distribution with parameters θ = 0.5 and α = 6 . (iii) The number of claims and claim sizes are independent. (iv) The observed pure premium should be within 2% of the expected pure premium 90% of the time. Determine the expected number of claims needed for full credibility. (A) Less than 7,000 (B) At least 7,000, but less than 10,000 (C) At least 10,000, but less than 13,000 (D) At least 13,000, but less than 16,000 (E) At least 16,000

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Course 4: Fall 2003 -4- GO ON TO NEXT PAGE 4. You study five lives to estimate the time from the onset of a disease to death. The times to death are: 2 3 3 3 7 Using a triangular kernel with bandwidth 2, estimate the density function at 2.5. (A) 8/40 (B) 12/40 (C) 14/40 (D) 16/40 (E) 17/40
Course 4: Fall 2003 -5- GO ON TO NEXT PAGE 5. For the model ii i YX α βε =+ + , where 1,2,. ..,10 i = , you are given: (i) X i i = R S T 1, if the th individual belongs to a specified group 0, otherwise (ii) 40 percent of the individuals belong to the specified group. (iii) The least squares estimate of β is ± = 4 . (iv) () 2 ˆ ˆ 92 αβ −− = Calculate the t statistic for testing H 0 0 : = . (A) 0.9 (B) 1.2 (C) 1.5 (D) 1.8 (E) 2.1

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Course 4: Fall 2003 -6- GO ON TO NEXT PAGE 6. You are given: (i) Losses follow a Single-parameter Pareto distribution with density function: () 1 ,1 f xx x α + = > , 0 < < (ii) A random sample of size five produced three losses with values 3, 6 and 14, and two losses exceeding 25. Determine the maximum likelihood estimate of . (A) 0.25 (B) 0.30 (C) 0.34 (D) 0.38 (E) 0.42
Course 4: Fall 2003 -7- GO ON TO NEXT PAGE 7. You are given: (i) The annual number of claims for a policyholder has a binomial distribution with probability function: () () 2 2 1 x x pxq q q x ⎛⎞ =− ⎜⎟ ⎝⎠ , x = 0, 1, 2 (ii) The prior distribution is: () 3 4, 0 1 qq q π = << This policyholder had one claim in each of Years 1 and 2. Determine the Bayesian estimate of the number of claims in Year 3. (A) Less than 1.1 (B) At least 1.1, but less than 1.3 (C) At least 1.3, but less than 1.5 (D) At least 1.5, but less than 1.7 (E) At least 1.7

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Course 4: Fall 2003 -8- GO ON TO NEXT PAGE 8. For a sample of dental claims 12 1 0 , ,..., x xx , you are given: (i) 2 3860 and 4,574,802 ii == ∑∑ (ii) Claims are assumed to follow a lognormal distribution with parameters µ and σ .
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course4_1103 - Course 4 Fall 2003 Society of...

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